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On conjectures of Olsson, Brauer, and Alperin

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Abstract

Let G be a finite group, p a prime number, B a p-block of the group G, k(B) the number of irreducible complex characters of R belonging to B, k0(B) the number of irreducible characters of height zero in B, and let D be the defect group of B. This article considers the relationship between Brauer's conjecture (k(B) ≤ ¦D¦), Olsson's conjecture (k0(B) ≤¦D/D'¦), and Alperin's conjecture (k0(B) = k0(~B), where ~B is a p-block NG(D) such that ~BG = B). In particular, Olsson's conjecture is proved for p-blocks for those p-solvable groups G for which a Hall p′-subgroup of the group NG(D) is either supersolvable or has odd order.

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Authors and Affiliations

  1. Uzhgorod State University, USSR

    P. G. Gres'

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  1. P. G. Gres'
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Translated from Matematicheskie Zametki, Vol. 52, No. 1, pp. 32–35, July, 1992.

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Gres', P.G. On conjectures of Olsson, Brauer, and Alperin. Math Notes 52, 654–657 (1992). https://doi.org/10.1007/BF01247644

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